**Mathematics Problem Set**1. **Revenue and Cost Analysis** If the total revenue received from the sale of $ x $ items is given by: \[ R(x) = 10 \ln(2x + 1) \] while the total cost to produce $ x $ items is: \[ C(x) = \frac{x}{2} \] Find the following: a) The marginal revenue b) The profit function $ P(x) $ c) The marginal profit when $ x = 60 $ d) Interpret the results of part c) 2. **Cost Function Analysis** Suppose the total cost $ C(x) $ (in dollars) to manufacture a quantity $ x $ of a weed killer (in hundreds of liters) is given by: \[ C(x) = x^3 - 2x^2 + 8x + 50 \] a) Where is $ C(x) $ increasing? b) Where is $ C(x) $ decreasing? 3. **Local Extrema Determination** Use the first derivative test to determine all local minima and maxima for the following function: \[ f(x) = 4x^3 - 9x^2 - 30x + 6 \]4. **Absolute Extrema on Closed Interval** Determine the absolute maximum and absolute minimum for the following function on the given closed interval: \[ f(x) = x^3 - 3x^2 - 24x + 5; \quad [-3, 6] \] ### Problem StatementA company has established that its weekly profit from selling $ x $ thousand hula hoops can be modeled by the equation:\[ P(x) = -0.02x^3 + 600x - 20,000 \]### Constraints- Production bottlenecks restrict the number of hula hoops that can be produced each week to a maximum of 150.- A long-term contract mandates a minimum production of 50 hula hoops each week.### ObjectiveDetermine the maximum possible weekly profit that the firm can achieve under these production constraints.### Solution Approach1. **Understand the Profit Function:** The profit $ P(x) $ is a cubic polynomial function of $ x $, representing the number of thousands of hula hoops sold.2. **Apply Constraints:** The constraints translate the problem domain to $ 50 \leq x \leq 150 $.3. **Maximize Profit:** Calculate the profit at the boundary values $ x = 50 $ and $ x = 150 $, and examine the turning points within the allowed range for global maxima.4. **Analyze:** Compare the profit values to find the maximum achievable profit within the given constraints.By applying the above approach, the company can strategize its production to maximize profits while adhering to production limitations.

As per the guidelines, I am providing the solution to the first question Answer 1: R(x) = 10…

1. If the total revenue received from the sale of x items is given by R(x) = 10 ln(2x + 1) while the total cost to produce x items is C(x) = a) The marginal revenue b) The profit function P(x) c) The marginal profit when x = 60 d) Interpret the results of part c) =. Find the following

1. If the total revenue received from the sale of x items is given by R(x) = 10 ln(2x + 1) while the total cost to produce x items is C(x) = a) The marginal revenue b) The profit function P(x) c) The marginal profit when x = 60 d) Interpret the results of part c) =. Find the following

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Mathematics Problem Set**

1. **Revenue and Cost Analysis**

If the total revenue received from the sale of $ x $ items is given by:
\[
R(x) = 10 \ln(2x + 1)
\]
while the total cost to produce $ x $ items is:
\[
C(x) = \frac{x}{2}
\]
Find the following:

a) The marginal revenue
b) The profit function $ P(x) $
c) The marginal profit when $ x = 60 $
d) Interpret the results of part c)

2. **Cost Function Analysis**

Suppose the total cost $ C(x) $ (in dollars) to manufacture a quantity $ x $ of a weed killer (in hundreds of liters) is given by:
\[
C(x) = x^3 - 2x^2 + 8x + 50
\]

a) Where is $ C(x) $ increasing?
b) Where is $ C(x) $ decreasing?

3. **Local Extrema Determination**

Use the first derivative test to determine all local minima and maxima for the following function:
\[
f(x) = 4x^3 - 9x^2 - 30x + 6
\]

4. **Absolute Extrema on Closed Interval**

Determine the absolute maximum and absolute minimum for the following function on the given closed interval:
\[
f(x) = x^3 - 3x^2 - 24x + 5; \quad [-3, 6]
\]

### Problem Statement

A company has established that its weekly profit from selling $ x $ thousand hula hoops can be modeled by the equation:

\[
P(x) = -0.02x^3 + 600x - 20,000
\]

### Constraints

- Production bottlenecks restrict the number of hula hoops that can be produced each week to a maximum of 150.
- A long-term contract mandates a minimum production of 50 hula hoops each week.

### Objective

Determine the maximum possible weekly profit that the firm can achieve under these production constraints.

### Solution Approach

1. **Understand the Profit Function:** The profit $ P(x) $ is a cubic polynomial function of $ x $, representing the number of thousands of hula hoops sold.

2. **Apply Constraints:** The constraints translate the problem domain to $ 50 \leq x \leq 150 $.

3. **Maximize Profit:** Calculate the profit at the boundary values $ x = 50 $ and $ x = 150 $, and examine the turning points within the allowed range for global maxima.

4. **Analyze:** Compare the profit values to find the maximum achievable profit within the given constraints.

By applying the above approach, the company can strategize its production to maximize profits while adhering to production limitations.

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